Integrand size = 21, antiderivative size = 137 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {21 a^4 x}{16}-\frac {7 a^4 \cos ^3(c+d x)}{8 d}+\frac {21 a^4 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}-\frac {21 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{40 d} \] Output:
21/16*a^4*x-7/8*a^4*cos(d*x+c)^3/d+21/16*a^4*cos(d*x+c)*sin(d*x+c)/d-1/6*a *cos(d*x+c)^3*(a+a*sin(d*x+c))^3/d-3/10*cos(d*x+c)^3*(a^2+a^2*sin(d*x+c))^ 2/d-21/40*cos(d*x+c)^3*(a^4+a^4*sin(d*x+c))/d
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {a^4 \cos ^3(c+d x) \left (630 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (448-373 \sin (c+d x)-331 \sin ^2(c+d x)-94 \sin ^3(c+d x)+158 \sin ^4(c+d x)+152 \sin ^5(c+d x)+40 \sin ^6(c+d x)\right )\right )}{240 d (-1+\sin (c+d x))^2 (1+\sin (c+d x))^{3/2}} \] Input:
Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]
Output:
-1/240*(a^4*Cos[c + d*x]^3*(630*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqr t[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(448 - 373*Sin[c + d*x] - 331 *Sin[c + d*x]^2 - 94*Sin[c + d*x]^3 + 158*Sin[c + d*x]^4 + 152*Sin[c + d*x ]^5 + 40*Sin[c + d*x]^6)))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/ 2))
Time = 0.66 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 3157, 3042, 3157, 3042, 3157, 3042, 3148, 3042, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^2(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^2 (a \sin (c+d x)+a)^4dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \int \cos ^2(c+d x) (\sin (c+d x) a+a)^3dx-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \int \cos (c+d x)^2 (\sin (c+d x) a+a)^3dx-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \int \cos ^2(c+d x) (\sin (c+d x) a+a)^2dx-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \int \cos (c+d x)^2 (\sin (c+d x) a+a)^2dx-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \int \cos ^2(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \int \cos (c+d x)^2 (\sin (c+d x) a+a)dx-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \int \cos ^2(c+d x)dx-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {a \cos ^3(c+d x)}{3 d}\right )-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{4 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}\right )-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}\) |
Input:
Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]
Output:
-1/6*(a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/d + (3*a*(-1/5*(a*Cos[c + d *x]^3*(a + a*Sin[c + d*x])^2)/d + (7*a*(-1/4*(Cos[c + d*x]^3*(a^2 + a^2*Si n[c + d*x]))/d + (5*a*(-1/3*(a*Cos[c + d*x]^3)/d + a*(x/2 + (Cos[c + d*x]* Sin[c + d*x])/(2*d))))/4))/5))/2
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Time = 118.99 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {\left (252 d x +\sin \left (6 d x +6 c \right )-288 \cos \left (d x +c \right )-80 \cos \left (3 d x +3 c \right )+\frac {48 \cos \left (5 d x +5 c \right )}{5}+45 \sin \left (2 d x +2 c \right )-39 \sin \left (4 d x +4 c \right )-\frac {1792}{5}\right ) a^{4}}{192 d}\) | \(76\) |
| risch | \(\frac {21 a^{4} x}{16}-\frac {3 a^{4} \cos \left (d x +c \right )}{2 d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} \cos \left (5 d x +5 c \right )}{20 d}-\frac {13 a^{4} \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 a^{4} \cos \left (3 d x +3 c \right )}{12 d}+\frac {15 a^{4} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
| derivativedivides | \(\frac {a^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+4 a^{4} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {4 a^{4} \cos \left (d x +c \right )^{3}}{3}+a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(182\) |
| default | \(\frac {a^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+4 a^{4} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {4 a^{4} \cos \left (d x +c \right )^{3}}{3}+a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(182\) |
| norman | \(\frac {\frac {21 a^{4} x}{16}-\frac {56 a^{4}}{15 d}-\frac {5 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {235 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {63 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {63 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}-\frac {235 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {5 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {63 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {315 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {105 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {315 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {63 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {21 a^{4} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}-\frac {8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {40 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {72 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {16 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {112 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(341\) |
| orering | \(\text {Expression too large to display}\) | \(3196\) |
Input:
int(cos(d*x+c)^2*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/192*(252*d*x+sin(6*d*x+6*c)-288*cos(d*x+c)-80*cos(3*d*x+3*c)+48/5*cos(5* d*x+5*c)+45*sin(2*d*x+2*c)-39*sin(4*d*x+4*c)-1792/5)*a^4/d
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {192 \, a^{4} \cos \left (d x + c\right )^{5} - 640 \, a^{4} \cos \left (d x + c\right )^{3} + 315 \, a^{4} d x + 5 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{5} - 86 \, a^{4} \cos \left (d x + c\right )^{3} + 63 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
1/240*(192*a^4*cos(d*x + c)^5 - 640*a^4*cos(d*x + c)^3 + 315*a^4*d*x + 5*( 8*a^4*cos(d*x + c)^5 - 86*a^4*cos(d*x + c)^3 + 63*a^4*cos(d*x + c))*sin(d* x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (128) = 256\).
Time = 0.39 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.78 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\begin {cases} \frac {a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {a^{4} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {4 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {8 a^{4} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {4 a^{4} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{4} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*(a+a*sin(d*x+c))**4,x)
Output:
Piecewise((a**4*x*sin(c + d*x)**6/16 + 3*a**4*x*sin(c + d*x)**4*cos(c + d* x)**2/16 + 3*a**4*x*sin(c + d*x)**4/4 + 3*a**4*x*sin(c + d*x)**2*cos(c + d *x)**4/16 + 3*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + a**4*x*sin(c + d* x)**2/2 + a**4*x*cos(c + d*x)**6/16 + 3*a**4*x*cos(c + d*x)**4/4 + a**4*x* cos(c + d*x)**2/2 + a**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) - a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 3*a**4*sin(c + d*x)**3*cos(c + d*x)/(4*d ) - 4*a**4*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 3*a**4*sin(c + d*x)*cos(c + d*x)**3/(4*d) + a**4*sin(c + d*x)*cos(c + d*x)/(2*d) - 8*a**4*cos(c + d*x)**5/(15*d) - 4*a**4*cos(c + d*x)**3/(3*d), Ne(d, 0)), (x*(a*sin(c) + a)**4*cos(c)**2, True))
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {1280 \, a^{4} \cos \left (d x + c\right )^{3} - 256 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{4} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 180 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{960 \, d} \] Input:
integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
-1/960*(1280*a^4*cos(d*x + c)^3 - 256*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3 )*a^4 + 5*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^4 - 180*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^4 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^4)/d
Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {21}{16} \, a^{4} x + \frac {a^{4} \cos \left (5 \, d x + 5 \, c\right )}{20 \, d} - \frac {5 \, a^{4} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {3 \, a^{4} \cos \left (d x + c\right )}{2 \, d} + \frac {a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {13 \, a^{4} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
21/16*a^4*x + 1/20*a^4*cos(5*d*x + 5*c)/d - 5/12*a^4*cos(3*d*x + 3*c)/d - 3/2*a^4*cos(d*x + c)/d + 1/192*a^4*sin(6*d*x + 6*c)/d - 13/64*a^4*sin(4*d* x + 4*c)/d + 15/64*a^4*sin(2*d*x + 2*c)/d
Time = 19.37 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.55 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {21\,a^4\,x}{16}-\frac {\frac {63\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-\frac {63\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {235\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {235\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-\frac {5\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {a^4\,\left (315\,c+315\,d\,x\right )}{240}-\frac {a^4\,\left (315\,c+315\,d\,x-896\right )}{240}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{40}-\frac {a^4\,\left (1890\,c+1890\,d\,x-1920\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{40}-\frac {a^4\,\left (1890\,c+1890\,d\,x-3456\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^4\,\left (4725\,c+4725\,d\,x-3840\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^4\,\left (4725\,c+4725\,d\,x-9600\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{12}-\frac {a^4\,\left (6300\,c+6300\,d\,x-8960\right )}{240}\right )+\frac {5\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int(cos(c + d*x)^2*(a + a*sin(c + d*x))^4,x)
Output:
(21*a^4*x)/16 - ((63*a^4*tan(c/2 + (d*x)/2)^7)/4 - (63*a^4*tan(c/2 + (d*x) /2)^5)/4 - (235*a^4*tan(c/2 + (d*x)/2)^3)/24 + (235*a^4*tan(c/2 + (d*x)/2) ^9)/24 - (5*a^4*tan(c/2 + (d*x)/2)^11)/8 + (a^4*(315*c + 315*d*x))/240 - ( a^4*(315*c + 315*d*x - 896))/240 + tan(c/2 + (d*x)/2)^10*((a^4*(315*c + 31 5*d*x))/40 - (a^4*(1890*c + 1890*d*x - 1920))/240) + tan(c/2 + (d*x)/2)^2* ((a^4*(315*c + 315*d*x))/40 - (a^4*(1890*c + 1890*d*x - 3456))/240) + tan( c/2 + (d*x)/2)^4*((a^4*(315*c + 315*d*x))/16 - (a^4*(4725*c + 4725*d*x - 3 840))/240) + tan(c/2 + (d*x)/2)^8*((a^4*(315*c + 315*d*x))/16 - (a^4*(4725 *c + 4725*d*x - 9600))/240) + tan(c/2 + (d*x)/2)^6*((a^4*(315*c + 315*d*x) )/12 - (a^4*(6300*c + 6300*d*x - 8960))/240) + (5*a^4*tan(c/2 + (d*x)/2))/ 8)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^6)
Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.73 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+350 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-75 \cos \left (d x +c \right ) \sin \left (d x +c \right )-448 \cos \left (d x +c \right )+315 d x +448\right )}{240 d} \] Input:
int(cos(d*x+c)^2*(a+a*sin(d*x+c))^4,x)
Output:
(a**4*(40*cos(c + d*x)*sin(c + d*x)**5 + 192*cos(c + d*x)*sin(c + d*x)**4 + 350*cos(c + d*x)*sin(c + d*x)**3 + 256*cos(c + d*x)*sin(c + d*x)**2 - 75 *cos(c + d*x)*sin(c + d*x) - 448*cos(c + d*x) + 315*d*x + 448))/(240*d)